• Corpus ID: 224713426

Super Cartan geometry and the super Ashtekar connection

@article{Eder2020SuperCG,
  title={Super Cartan geometry and the super Ashtekar connection},
  author={Konstantin Eder},
  journal={arXiv: General Relativity and Quantum Cosmology},
  year={2020}
}

• K. Eder
• Published 19 October 2020
• Physics
• arXiv: General Relativity and Quantum Cosmology

This work is devoted to the geometric approach to supergravity. More precisely, we interpret N = 1 supergravity as a super Cartan geometry which provides a link between supergravity and super Yang-Mills theory. To this end, we first review important aspects of the theory of supermanifolds and we establish a link between various different approaches. We will then use this gauge-theoretic interpretation and the methods of loop quantum gravity to canonically quantize this theory which preserves some of the underyling supersymmetry in a manifest way. More precisely, studying the chiral structure of the underlying supersymmetry algebra, we derive a graded analog of Ashtekar's self-dual variables and interpret them in terms of generalized super Cartan connections. This gives canonical supergravity the structure of a super Yang-Mills theory similar to the self-dual variables in ordinary first-order Einstein gravity which was first observed by F\"ul\"op. We then construct the parallel transport map corresponding to the super connection in mathematical rigorous way using the concept of enriched categories. This then provides the possibility of quantizing gravity and matter degrees of freedom in loop quantum gravity in a unified way. Finally, we study spatially symmetry reduced models and discuss possible applications of this approach in supersymmetric loop quantum cosmology.